Amplitude Analysis and Measurement of the Time- >K[subscript S][superscript 0]K[subscript S][superscript

Amplitude Analysis and Measurement of the Timedependent CP Asymmetry of B[superscript 0]->K[subscript S][superscript 0]K[subscript S][superscript The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Lees, J. et al. “Amplitude analysis and measurement of the timedependent CP asymmetry of B0KS0KS0KS0 decays.” Physical Review D 85.5 (2012): 054023-1-054023-21. © 2012 American Physical Society. As Published http://dx.doi.org/10.1103/PhysRevD.85.054023 Publisher American Physical Society Version Final published version Accessed Fri May 27 00:50:03 EDT 2016 Citable Link http://hdl.handle.net/1721.1/73117 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Detailed Terms PHYSICAL REVIEW D 85, 054023 (2012) Amplitude analysis and measurement of the time-dependent CP asymmetry of B0 ! KS0 KS0 KS0 decays J. P. Lees,1 V. Poireau,1 V. Tisserand,1 J. Garra Tico,2 E. Grauges,2 M. Martinelli,3a,3b D. A. Milanes,3a A. Palano,3a,3b M. Pappagallo,3a,3b G. Eigen,4 B. Stugu,4 D. N. Brown,5 L. T. Kerth,5 Yu. G. Kolomensky,5 G. Lynch,5 H. Koch,6 T. Schroeder,6 D. J. Asgeirsson,7 C. Hearty,7 T. S. Mattison,7 J. A. McKenna,7 A. Khan,8 V. E. Blinov,9 A. R. Buzykaev,9 V. P. Druzhinin,9 V. B. Golubev,9 E. A. Kravchenko,9 A. P. Onuchin,9 S. I. Serednyakov,9 Yu. I. Skovpen,9 E. P. Solodov,9 K. Yu. Todyshev,9 A. N. Yushkov,9 M. Bondioli,10 D. Kirkby,10 A. J. Lankford,10 M. Mandelkern,10 D. P. Stoker,10 H. Atmacan,11 J. W. Gary,11 F. Liu,11 O. Long,11 G. M. Vitug,11 C. Campagnari,12 T. M. Hong,12 D. Kovalskyi,12 J. D. Richman,12 C. A. West,12 A. M. Eisner,13 J. Kroseberg,13 W. S. Lockman,13 A. J. Martinez,13 T. Schalk,13 B. A. Schumm,13 A. Seiden,13 C. H. Cheng,14 D. A. Doll,14 B. Echenard,14 K. T. Flood,14 D. G. Hitlin,14 P. Ongmongkolkul,14 F. C. Porter,14 A. Y. Rakitin,14 R. Andreassen,15 M. S. Dubrovin,15 Z. Huard,15 B. T. Meadows,15 M. D. Sokoloff,15 L. Sun,15 P. C. Bloom,16 W. T. Ford,16 A. Gaz,16 M. Nagel,16 U. Nauenberg,16 J. G. Smith,16 S. R. Wagner,16 R. Ayad,17,* W. H. Toki,17 B. Spaan,18 M. J. Kobel,19 K. R. Schubert,19 R. Schwierz,19 D. Bernard,20 M. Verderi,20 P. J. Clark,21 S. Playfer,21 D. Bettoni,22a C. Bozzi,22a R. Calabrese,22a,22b G. Cibinetto,22a,22b E. Fioravanti,22a,22b I. Garzia,22a,22b E. Luppi,22a,22b M. Munerato,22a,22b M. Negrini,22a,22b L. Piemontese,22a V. Santoro,22a R. Baldini-Ferroli,23 A. Calcaterra,23 R. de Sangro,23 G. Finocchiaro,23 M. Nicolaci,23 P. Patteri,23 I. M. Peruzzi,23,† M. Piccolo,23 M. Rama,23 A. Zallo,23 R. Contri,24a,24b E. Guido,24a,24b M. Lo Vetere,24a,24b M. R. Monge,24a,24b S. Passaggio,24a C. Patrignani,24a,24b E. Robutti,24a B. Bhuyan,25 V. Prasad,25 C. L. Lee,26 M. Morii,26 A. J. Edwards,27 A. Adametz,28 J. Marks,28 U. Uwer,28 F. U. Bernlochner,29 M. Ebert,29 H. M. Lacker,29 T. Lueck,29 P. D. Dauncey,30 M. Tibbetts,30 P. K. Behera,31 U. Mallik,31 C. Chen,32 J. Cochran,32 W. T. Meyer,32 S. Prell,32 E. I. Rosenberg,32 A. E. Rubin,32 A. V. Gritsan,33 Z. J. Guo,33 N. Arnaud,34 M. Davier,34 G. Grosdidier,34 F. Le Diberder,34 A. M. Lutz,34 B. Malaescu,34 P. Roudeau,34 M. H. Schune,34 A. Stocchi,34 G. Wormser,34 D. J. Lange,35 D. M. Wright,35 I. Bingham,36 C. A. Chavez,36 J. P. Coleman,36 J. R. Fry,36 E. Gabathuler,36 D. E. Hutchcroft,36 D. J. Payne,36 C. Touramanis,36 A. J. Bevan,37 F. Di Lodovico,37 R. Sacco,37 M. Sigamani,37 G. Cowan,38 D. N. Brown,39 C. L. Davis,39 A. G. Denig,40 M. Fritsch,40 W. Gradl,40 A. Hafner,40 E. Prencipe,40 K. E. Alwyn,41 D. Bailey,41 R. J. Barlow,41,‡ G. Jackson,41 G. D. Lafferty,41 E. Behn,42 R. Cenci,42 B. Hamilton,42 A. Jawahery,42 D. A. Roberts,42 G. Simi,42 C. Dallapiccola,43 R. Cowan,44 D. Dujmic,44 G. Sciolla,44 D. Lindemann,45 P. M. Patel,45 S. H. Robertson,45 M. Schram,45 P. Biassoni,46a,46b A. Lazzaro,46a,46b V. Lombardo,46a N. Neri,46a,46b F. Palombo,46a,46b S. Stracka,46a,46b L. Cremaldi,47 R. Godang,47,§ R. Kroeger,47 P. Sonnek,47 D. J. Summers,47 X. Nguyen,48 P. Taras,48 G. De Nardo,49a,49b D. Monorchio,49a,49b G. Onorato,49a,49b C. Sciacca,49a,49b G. Raven,50 H. L. Snoek,50 C. P. Jessop,51 K. J. Knoepfel,51 J. M. LoSecco,51 W. F. Wang,51 K. Honscheid,52 R. Kass,52 J. Brau,53 R. Frey,53 N. B. Sinev,53 D. Strom,53 E. Torrence,53 E. Feltresi,54a,54b N. Gagliardi,54a,54b M. Margoni,54a,54b M. Morandin,54a M. Posocco,54a M. Rotondo,54a F. Simonetto,54a,54b R. Stroili,54a,54b S. Akar,55 E. Ben-Haim,55 M. Bomben,55 G. R. Bonneaud,55 H. Briand,55 G. Calderini,55 J. Chauveau,55 O. Hamon,55 Ph. Leruste,55 G. Marchiori,55 J. Ocariz,55 S. Sitt,55 M. Biasini,56a,56b E. Manoni,56a,56b S. Pacetti,56a,56b A. Rossi,56a,56b C. Angelini,57a,57b G. Batignani,57a,57b S. Bettarini,57a,57b M. Carpinelli,57a,57b,k G. Casarosa,57a,57b A. Cervelli,57a,57b F. Forti,57a,57b M. A. Giorgi,57a,57b A. Lusiani,57a,57c B. Oberhof,57a,57b E. Paoloni,57a,57b A. Perez,57a G. Rizzo,57a,57b J. J. Walsh,57a D. Lopes Pegna,58 C. Lu,58 J. Olsen,58 A. J. S. Smith,58 A. V. Telnov,58 F. Anulli,59a G. Cavoto,59a R. Faccini,59a,59b F. Ferrarotto,59a F. Ferroni,59a,59b L. Li Gioi,59a M. A. Mazzoni,59a G. Piredda,59a C. Bünger,60 O. Grünberg,60 T. Hartmann,60 T. Leddig,60 H. Schröder,60 R. Waldi,60 T. Adye,61 E. O. Olaiya,61 F. F. Wilson,61 S. Emery,62 G. Hamel de Monchenault,62 G. Vasseur,62 Ch. Yèche,62 D. Aston,63 D. J. Bard,63 R. Bartoldus,63 C. Cartaro,63 M. R. Convery,63 J. Dorfan,63 G. P. Dubois-Felsmann,63 W. Dunwoodie,63 R. C. Field,63 M. Franco Sevilla,63 B. G. Fulsom,63 A. M. Gabareen,63 M. T. Graham,63 P. Grenier,63 C. Hast,63 W. R. Innes,63 M. H. Kelsey,63 H. Kim,63 P. Kim,63 M. L. Kocian,63 D. W. G. S. Leith,63 P. Lewis,63 S. Li,63 B. Lindquist,63 S. Luitz,63 V. Luth,63 H. L. Lynch,63 D. B. MacFarlane,63 D. R. Muller,63 H. Neal,63 S. Nelson,63 I. Ofte,63 M. Perl,63 T. Pulliam,63 B. N. Ratcliff,63 A. Roodman,63 A. A. Salnikov,63 R. H. Schindler,63 A. Snyder,63 D. Su,63 M. K. Sullivan,63 J. Va’vra,63 A. P. Wagner,63 M. Weaver,63 W. J. Wisniewski,63 M. Wittgen,63 D. H. Wright,63 H. W. Wulsin,63 A. K. Yarritu,63 C. C. Young,63 V. Ziegler,63 W. Park,64 M. V. Purohit,64 R. M. White,64 J. R. Wilson,64 A. Randle-Conde,65 S. J. Sekula,65 M. Bellis,66 J. F. Benitez,66 P. R. Burchat,66 T. S. Miyashita,66 M. S. Alam,67 J. A. Ernst,67 R. Gorodeisky,68 N. Guttman,68 D. R. Peimer,68 A. Soffer,68 P. Lund,69 S. M. Spanier,69 R. Eckmann,70 J. L. Ritchie,70 A. M. Ruland,70 C. J. Schilling,70 R. F. Schwitters,70 B. C. Wray,70 J. M. Izen,71 X. C. Lou,71 F. Bianchi,72a,72b D. Gamba,72a,72b L. Lanceri,73a,73b 1550-7998= 2012=85(5)=054023(21) 054023-1 Ó 2012 American Physical Society J. P. LEES et al. 73a,73b PHYSICAL REVIEW D 85, 054023 (2012) 74 74 75 L. Vitale, F. Martinez-Vidal, A. Oyanguren, H. Ahmed, J. Albert,75 Sw. Banerjee,75 H. H. F. Choi,75 G. J. King,75 R. Kowalewski,75 M. J. Lewczuk,75 I. M. Nugent,75 J. M. Roney,75 R. J. Sobie,75 N. Tasneem,75 T. J. Gershon,76 P. F. Harrison,76 T. E. Latham,76 E. M. T. Puccio,76 H. R. Band,77 S. Dasu,77 Y. Pan,77 R. Prepost,77 and S. L. Wu77 (BABAR Collaboration) 1 Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Université de Savoie, CNRS/IN2P3, F-74941 Annecy-Le-Vieux, France 2 Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain 3a INFN Sezione di Bari, I-70126 Bari, Italy; 3b Dipartimento di Fisica, Università di Bari, I-70126 Bari, Italy 4 University of Bergen, Institute of Physics, N-5007 Bergen, Norway 5 Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA 6 Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany 7 University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1 8 Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom 9 Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia 10 University of California at Irvine, Irvine, California 92697, USA 11 University of California at Riverside, Riverside, California 92521, USA 12 University of California at Santa Barbara, Santa Barbara, California 93106, USA 13 University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA 14 California Institute of Technology, Pasadena, California 91125, USA 15 University of Cincinnati, Cincinnati, Ohio 45221, USA 16 University of Colorado, Boulder, Colorado 80309, USA 17 Colorado State University, Fort Collins, Colorado 80523, USA 18 Technische Universität Dortmund, Fakultät Physik, D-44221 Dortmund, Germany 19 Technische Universität Dresden, Institut für Kern- und Teilchenphysik, D-01062 Dresden, Germany 20 Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France 21 University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom 22a INFN Sezione di Ferrara, I-44100 Ferrara, Italy; 22b Dipartimento di Fisica, Università di Ferrara, I-44100 Ferrara, Italy 23 INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy 24a INFN Sezione di Genova, I-16146 Genova, Italy; 24b Dipartimento di Fisica, Università di Genova, I-16146 Genova, Italy 25 Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India 26 Harvard University, Cambridge, Massachusetts 02138, USA 27 Harvey Mudd College, Claremont, California 91711, USA 28 Universität Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany 29 Humboldt-Universität zu Berlin, Institut für Physik, Newtonstrasse 15, D-12489 Berlin, Germany 30 Imperial College London, London, SW7 2AZ, United Kingdom 31 University of Iowa, Iowa City, Iowa 52242, USA 32 Iowa State University, Ames, Iowa 50011-3160, USA 33 Johns Hopkins University, Baltimore, Maryland 21218, USA 34 Laboratoire de l’Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11, Centre Scientifique d’Orsay, Boite Postale 34, F-91898 Orsay Cedex, France 35 Lawrence Livermore National Laboratory, Livermore, California 94550, USA 36 University of Liverpool, Liverpool L69 7ZE, United Kingdom 37 Queen Mary, University of London, London, E1 4NS, United Kingdom 38 University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom 39 University of Louisville, Louisville, Kentucky 40292, USA 40 Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany 41 University of Manchester, Manchester M13 9PL, United Kingdom 42 University of Maryland, College Park, Maryland 20742, USA 43 University of Massachusetts, Amherst, Massachusetts 01003, USA 44 Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA 45 McGill University, Montréal, Québec, Canada H3A 2T8 46a INFN Sezione di Milano, I-20133 Milano, Italy; 46b Dipartimento di Fisica, Università di Milano, I-20133 Milano, Italy 054023-2 AMPLITUDE ANALYSIS AND MEASUREMENT OF THE . . . PHYSICAL REVIEW D 85, 054023 (2012) 47 University of Mississippi, University, Mississippi 38677, USA Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7 49a INFN Sezione di Napoli, I-80126 Napoli, Italy; 49b Dipartimento di Scienze Fisiche, Università di Napoli Federico II, I-80126 Napoli, Italy 50 NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands 51 University of Notre Dame, Notre Dame, Indiana 46556, USA 52 Ohio State University, Columbus, Ohio 43210, USA 53 University of Oregon, Eugene, Oregon 97403, USA 54a INFN Sezione di Padova, I-35131 Padova, Italy; 54b Dipartimento di Fisica, Università di Padova, I-35131 Padova, Italy 55 Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France 56a INFN Sezione di Perugia, I-06100 Perugia, Italy; 56b Dipartimento di Fisica, Università di Perugia, I-06100 Perugia, Italy 57a INFN Sezione di Pisa, I-56127 Pisa, Italy; 57b Dipartimento di Fisica, Università di Pisa, I-56127 Pisa, Italy; 57c Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy 58 Princeton University, Princeton, New Jersey 08544, USA 59a INFN Sezione di Roma, I-00185 Roma, Italy; 59b Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Roma, Italy 60 Universität Rostock, D-18051 Rostock, Germany 61 Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom 62 CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France 63 SLAC National Accelerator Laboratory, Stanford, California 94309 USA 64 University of South Carolina, Columbia, South Carolina 29208, USA 65 Southern Methodist University, Dallas, Texas 75275, USA 66 Stanford University, Stanford, California 94305-4060, USA 67 State University of New York, Albany, New York 12222, USA 68 Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel 69 University of Tennessee, Knoxville, Tennessee 37996, USA 70 University of Texas at Austin, Austin, Texas 78712, USA 71 University of Texas at Dallas, Richardson, Texas 75083, USA 72a INFN Sezione di Torino, I-10125 Torino, Italy; 72b Dipartimento di Fisica Sperimentale, Università di Torino, I-10125 Torino, Italy 73a INFN Sezione di Trieste, I-34127 Trieste, Italy; 73b Dipartimento di Fisica, Università di Trieste, I-34127 Trieste, Italy 74 IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain 75 University of Victoria, Victoria, British Columbia, Canada V8W 3P6 76 Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom 77 University of Wisconsin, Madison, Wisconsin 53706, USA (Received 16 November 2011; published 27 March 2012) 48 We present the first results on the Dalitz-plot structure and improved measurements of the timedependent CP-violation parameters of the process B0 ! KS0 KS0 KS0 obtained using 468 106 BB decays collected with the BABAR detector at the PEP-II asymmetric-energy B factory at SLAC. The Dalitz-plot structure is probed by a time-integrated amplitude analysis that does not distinguish between B0 and B 0 decays. We measure the total inclusive branching fraction BðB0 ! KS0 KS0 KS0 Þ ¼ ð6:19 0:48 0:15 0:12Þ 106 , where the first uncertainty is statistical, the second is systematic, and the third represents the Dalitz-plot signal model dependence. We also observe evidence for the intermediate resonant states f0 ð980Þ, f0 ð1710Þ, and f2 ð2010Þ. Their respective product branching fractions are measured to be 6 þ0:46 6 þ0:21 ð2:70þ1:25 1:19 0:36 1:17Þ 10 , ð0:500:24 0:04 0:10Þ 10 , and ð0:540:20 0:03 0:52Þ 6 10 . Additionally, we determine the mixing-induced CP-violation parameters to be S ¼ 0:94þ0:24 0:21 0:06 and C ¼ 0:17 0:18 0:04, where the first uncertainty is statistical and the second is systematic. These values are in agreement with the standard model expectation. For the first time, we report evidence *Now at Temple University, Philadelphia, PA 19122, USA. † Also with Università di Perugia, Dipartimento di Fisica, Perugia, Italy. ‡ Now at the University of Huddersfield, Huddersfield HD1 3DH, UK. § Now at University of South Alabama, Mobile, AL 36688, USA. k Also with Università di Sassari, Sassari, Italy. 054023-3 J. P. LEES et al. PHYSICAL REVIEW D 85, 054023 (2012) of CP violation in B ! including systematic uncertainties. 0 KS0 KS0 KS0 decays; CP conservation is excluded at 3.8 standard deviations DOI: 10.1103/PhysRevD.85.054023 PACS numbers: 13.66.Bc, 13.25.Gv, 13.25.Jx, 14.40.n I. INTRODUCTION Over the past ten years, the B factories have shown that the Cabibbo-Kobayashi-Maskawa paradigm in the standard model (SM), with a single weak phase in the quark mixing matrix, accounts for the observed CP-symmetry violation in the quark sector. However, there may be other CP-violating sources beyond the SM. Charmless hadronic B decays, like B0 ! KS0 KS0 KS0 , are of great interest because they are dominated by loop diagrams and are thus sensitive to new physics effects at large energy scales [1]. In the SM, the mixing-induced CP-violation parameters in this decay are expected to be the same, up to 1% [2], as in the treediagram-dominated modes such as B0 ! J= c KS0 . Both BABAR [3] and Belle [4] have previously performed time-dependent CP-violation measurements of the inclusive mode B0 ! KS0 KS0 KS0 , which is permissible because the final state is CP definite [5]. The structure of the Dalitz plot (DP), however, is of interest; although the time-dependent CP-violation parameters S and C [see Eq. (33)] can be measured inclusively without taking into account the phase space, different resonant contributions may have different values of these parameters in the presence of new physics. The statistical precision is not sufficient to perform a timedependent amplitude analysis, but as we show below, it is possible to extract branching fractions from resonant contributions to the decay using a time-integrated amplitude analysis. Additionally, the amplitude analysis could shed light on the controversial fX ð1500Þ resonance: recent measurements of B0 ! Kþ K KS0 and B ! Kþ K K from BABAR [6–8] and Belle [9,10] have shown evidence of a wide structure in the mKþ K spectrum around 1.5 GeV. In these measurements, it was assumed that this structure is a single scalar resonance; however, a vector hypothesis could not be ruled out. The BABAR measurement of Bþ ! K þ K þ [11] appears to show an enhancement around 1.5 GeV, while the BABAR analysis of B ! KS0 KS0 [12] finds no evidence of a possible fX ð1500Þ, suggesting that the structure is either a vector meson or something exotic. An amplitude analysis of B0 ! KS0 KS0 KS0 will provide further insight into the nature of this structure, as only intermediate states of even spin are permitted due to BoseEinstein statistics; an observation of the fX ð1500Þ decaying to KS0 KS0 would require an even-spin state. Finally, the amplitude analyses of B ! K and B ! KKK modes may be used to extract the Cabibbo-Kobayashi-Maskawa angle [13]. This paper presents the first amplitude analysis and the final BABAR update of the time-dependent CP-asymmetry measurement of B0 ! KS0 KS0 KS0 using the full ð4SÞ data set. The amplitude analysis is time-integrated CP averaged (i.e., it does not use flavor-tagging information to distinguish between B0 and B 0 mesons). It takes advantage of the interference pattern in the DP to measure relative magnitudes and phases for the different resonant modes using B0 ! KS0 KS0 KS0 decays with KS0 ! þ , denoted by B0 ! 3KS0 ðþ Þ. The magnitudes and phases are then translated into individual branching fractions for the resonant modes. The time-dependent analysis extracts the S and C parameters by modeling the proper-time distribution. This part of the analysis uses both B0 ! 3KS0 ðþ Þ events and events where one of the KS0 mesons decays to 0 0 , denoted by B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ. In Sec. II we briefly describe the BABAR detector and the data set. The amplitude analysis is described in Sec. III and the time-dependent analysis in Sec. IV. Finally we summarize the results in Sec. V. II. THE BABAR DETECTOR AND DATA SET The data used in this analysis were collected with the BABAR detector at the PEP-II asymmetric-energy eþ e storage ring at SLAC. The sample consists of an integrated luminosity of 426:0 fb1 , corresponding to ð467:8 5:1Þ 106 BB pairs collected at the ð4SÞ resonance (‘‘on-resonance’’), and 44:5 fb1 collected about 40 MeV below the ð4SÞ (‘‘off-resonance’’). A detailed description of the BABAR detector is presented in Ref. [14]. The tracking system used for track and vertex reconstruction has two components: a silicon vertex tracker and a drift chamber, both operating within a 1.5 T magnetic field generated by a superconducting solenoidal magnet. A detector of internally reflected Cherenkov light associates Cherenkov photons with tracks for particle identification. The energies of photons and electrons are determined from the measured light produced in electromagnetic showers inside a CsI crystal electromagnetic calorimeter. Muon candidates are identified with the use of the instrumented flux return of the solenoid. III. AMPLITUDE ANALYSIS In Secs. III A and III B we describe the DP formalism and introduce the signal parameters that are extracted from data. In Sec. III C we describe the requirements used to select the signal candidates and suppress backgrounds. In Sec. III D we describe the fit method and the approach used to account for experimental effects such as resolution. In Sec. III E we present the results of the fit, and finally, in Sec. III F we discuss systematic uncertainties in the results. 054023-4 AMPLITUDE ANALYSIS AND MEASUREMENT OF THE . . . PHYSICAL REVIEW D 85, 054023 (2012) A. Decay amplitudes decay contains three identical parThe B ! ticles in the final state and therefore the amplitude needs to be symmetrized. We consider the decay of a spin-zero B0 into three daughters, KS0 ð1Þ, KS0 ð2Þ, and KS0 ð3Þ, with fourmomenta p1 , p2 , and p3 . The decay amplitude is given by [2] 0 A ðsmin ; smax Þ ¼ KS0 KS0 KS0 A½B0 ! KS0 ð1ÞKS0 ð2ÞKS0 ð3Þ ¼ ð12Þ3=2 fA1 ½B0 ! K 0 ð1ÞK0 ð2ÞK0 ð3Þ þ A2 ½B0 ! K 0 ð2ÞK0 ð3ÞK0 ð1Þ þ A3 ½B0 ! K 0 ð3ÞK0 ð1ÞK0 ð2Þg; (1) which takes into account the three permitted paths from the initial state to the final state. For instance for the B0 decay this consists of an intermediate state K0 K0 K 0 . Since the labeling of the three identical particles is arbitrary, we classify the final-state particles according to the square of the invariant mass, sij , defined as sij ¼ sji ¼ m2K0 ðiÞK0 ðjÞ ¼ ðpi þ pj Þ2 ; S where i and j are the KS0 indices. We use as independent (Mandelstam) variables the minimum and the maximum of the squared masses smin and smax : smin ¼ minðs12 ; s23 ; s13 Þ; (3) smax ¼ maxðs12 ; s23 ; s13 Þ: The third (median) invariant squared mass smed can be obtained from energy and momentum conservation: smed ¼ m2B0 þ 3m2K0 smin smax : (4) S The differential B meson decay width with respect to the variables defined in Eq. (3) (i.e., the DP variables) reads dðB ! KS0 KS0 KS0 Þ ¼ 1 jAj2 dsmin dsmax ; ð2Þ3 32m3B0 (5) where A is the Lorentz-invariant amplitude of the threebody decay. This amplitude analysis does not take into account any flavor tagging or time dependence; thus it is CP averaged and time integrated. The term jAj2 is therefore simply the average of squares of the contributions A½B0 ! KS0 KS0 KS0 and A½B 0 ! KS0 KS0 KS0 . The choice of the variables smin and smax gives a uniquely defined coordinate in the symmetrized DP. Therefore only one-sixth of the DP is populated; i.e., the event density is 6 times larger compared to an amplitude analysis involving three distinct particles. We describe the distribution of signal events in the DP using an isobar approximation, which models the total amplitude as resulting from a coherent sum of amplitudes from the N individual decay channels of the B meson, either into an intermediate resonance and a bachelor particle or in a nonresonant manner: cj Fj ðsmin ; smax Þ: (6) j¼1 Here Fj (described in detail below) are DP-dependent amplitudes containing the decay dynamics and cj are complex coefficients describing the relative magnitudes and phases of the different decay channels. This description, which contains a single complex number cj for each decay channel regardless of the B- flavor (B0 or B 0 ), reflects the assumptions of no direct CP violation and of a common weak phase for all the decay channels. With this description we cannot extract any weak phase information; this would require using per-B flavor complex amplitudes. The factor Fj contains strong dynamics only, and thus does not change under CP conjugation. Intermediate resonances decay to K0 K 0 . In terms of the isobar approximation, the amplitude in Eq. (1) for a resonant state j becomes A ½B0 ! KS0 ð1ÞKS0 ð2ÞKS0 ð3Þ / cj ½Fj ðs12 ;s13 Þ þ Fj ðs12 ;s23 Þ þ Fj ðs13 ;s23 Þ: (7) (2) S N X This reflects the fact that it is impossible to associate a given KS0 to a flavor eigenstate K 0 or K 0 . In practice, this sum of three Fj terms, corresponding to an even-spin resonance, is implicitly taken into account by the description in terms of smin and smax . The Fj terms are represented by the product of the invariant mass and angular distributions; i.e., ~ ~ qÞ; ~ Fj ðsmin ; smax ; LÞ ¼ Rj ðmÞXL ðjp~ ? jr0 ÞXL ðjqjrÞT j ðL; p; (8) where (i) m is the invariant mass of the decay products of the resonance, (ii) Rj ðmÞ is the resonance mass term or ‘‘line shape,’’ e.g., relativistic Breit-Wigner (RBW), (iii) L is the orbital angular momentum between the resonance and the bachelor particle, (iv) p~ ? is the momentum of the bachelor particle evaluated in the rest frame of the B, (v) p~ and q~ are the momenta of the bachelor particle and one of the resonance daughters, respectively, both evaluated in the rest frame of the resonance, ~ are Blatt-Weisskopf barrier (vi) XL ðjp~ ? jr0 Þ and XL ðjqjrÞ factors [15] with barrier radii of r and r0 , and ~ qÞ ~ is the angular distribution: (vii) Tj ðL; p; L ¼ 0: Tj ¼ 1; ~ 2 ðjpjj ~ qjÞ ~ 2 : L ¼ 2: Tj ¼ 83½3ðp~ qÞ (9) (10) The Blatt-Weisskopf barrier factor is unity for all the zerospin resonances. In our analysis it is relevant only for the f2 ð2010Þ. Since for this resonance r and r0 are not measured, we take them both to be 1:5 GeV1 and vary by 0:5 GeV1 to estimate the systematic uncertainty. 054023-5 J. P. LEES et al. PHYSICAL REVIEW D 85, 054023 (2012) The helicity angle of a resonance is defined as the angle ~ Explicitly, the helicity angle for a given between p~ and q. resonance is defined between the momenta of the bachelor particle and one of the daughters of the resonance in the resonance rest frame. Because of the identical final-state particles this definition is ambiguous, but the ambiguity disappears because of the description of the DP in terms of smin and smax . There are three possible invariant-mass combinations: smin , smed , and smax . We denote the corresponding helicity angles as min , med , and max . The three angles are defined between 0 and =2. As the present study is the first amplitude analysis of this decay, we use the method outlined in Sec. III D 3 to determine the contributing intermediate states. The components of the nominal signal model are summarized in Table I. For most resonances in this analysis the Rj are taken to be RBW [17] line shapes: Rj ðmÞ ¼ 1 ; ðm20 m2 Þ im0 ðmÞ (11) where m0 is the nominal mass of the resonance and ðmÞ is the mass-dependent width. In the general case of a spin-J resonance, the latter can be expressed as 2Jþ1 2 ~ q m0 XJ ðjqjrÞ ðmÞ ¼ 0 : (12) q0 m XJ2 ðjq~ 0 jrÞ The symbol 0 denotes the nominal width of the resonance. The values of m0 and 0 are listed in Table I. The symbol q0 denotes the value of q when m ¼ m0 . For the f0 ð980Þ line shape the Flatté form [18] is used. In this case the mass-dependent width is given by the sum of the widths in the and KK systems: ðmÞ ¼ ðmÞ þ KK ðmÞ; (13) where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðmÞ ¼ g ð13 1 4m20 =m2 þ 23 1 4m2 =m2 ; (14) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi KK ðmÞ ¼ gK ð12 1 4m2K =m2 þ 12 1 4m2K0 =m2 Þ: (15) The fractional coefficients arise from isospin conservation and g and gK are coupling constants for which the values are given in Table I. The nonresonant (NR) component is modeled using an exponential function: RNR ðmÞ ¼ em : 2 As in the resonant case, here m is the invariant mass of the relevant KS0 KS0 pair. The parameter is taken from the BABAR Bþ ! K þ K K þ analysis [7,8] and is given in Table I. This value was found to be compatible with the one resulting from varying in the maximum-likelihood fit in the present analysis. There is no satisfactory theoretical description of the NR component; it has to be determined empirically. The exponential function of Eq. (16) was used by other amplitude analyses of B-meson decays to three kaons [6–10]. Adopting the same parametrization for the NR term allows the comparison of results for other components. B. The square Dalitz plot We use two-dimensional histograms to describe the phase-space dependent reconstruction efficiency and to model the background over the DP. When the phase-space boundaries of the DP do not coincide with the histogram bin boundaries this may introduce biases. We therefore define hmin and hmax as cosmin and cosmax , respectively, and apply the transformation TABLE I. Parameters of the DP model used in the fit. The Blatt-Weisskopf barrier parameters (r and r0 ) of the f2 ð2010Þ, which have not been measured, are varied by 0:5 GeV1 for the model uncertainty. Resonance Parameters Line shape Reference f0 ð980Þ MeV=c2 m0 ¼ ð965 10Þ g ¼ ð165 18Þ MeV=c2 gK ¼ ð695 93Þ MeV=c2 Flatté Eq. (13) [16] f0 ð1710Þ m0 ¼ ð1724 7Þ MeV=c2 0 ¼ ð137 8Þ MeV=c2 RBW Eq. (11) [17] f2 ð2010Þ 2 m0 ¼ ð2011þ60 80 Þ MeV=c 0 ¼ ð202 60Þ MeV=c2 r ¼ r0 ¼ 1:5 GeV1 RBW Eq. (11) [17] ¼ ð0:14 0:02Þ GeV2 c4 Exponential NR Eq. (16) [8] m0 ¼ ð3414:75 0:31Þ MeV=c2 0 ¼ ð10:2 0:7Þ MeV=c2 RBW Eq. (11) [17] NR decays c0 (16) 054023-6 AMPLITUDE ANALYSIS AND MEASUREMENT OF THE . . . ðsmin ; smax Þ ! ðhmin ; hmax Þ: (17) The ðhmin ; hmax Þ plane is referred to as the square Dalitz plot (SDP), where both hmin and hmax range between 0 and 1 due to the convention adopted for the helicity angles (see Fig. 1). Explicitly, the transformation is s ðsmax smed Þ hmin ¼ qmin ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2min 4m2K0 smin S 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 ðmB0 mK0 smin Þ2 4m2K0 smin S (18) S smax ðsmed smin Þ hmax ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s2max 4m2K0 smax S 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðm2B0 m2K0 smax Þ2 4m2K0 smax S (19) S where the numerators may easily be expressed in terms of smin and smax using Eq. (4). The differential surface elements of the DP and the SDP are related by dsmin dsmax ¼ j detJjdhmin dhmax ; (20) where J ¼ Jðhmin ; hmax Þ is the appropriate Jacobian matrix. The backward transformations smin ðhmin ; hmax Þ and smax ðhmin ; hmax Þ, and therefore the Jacobian j detJj, cannot be found analytically; they are obtained numerically. The variables hmin and hmax as a function of the invariant masses are shown in Fig. 1 together with the Jacobian. C. Event selection and backgrounds We reconstruct B0 ! KS0 KS0 KS0 candidates from three ! þ candidates that form a good quality vertex; i.e., the fit of the B0 vertex is required to converge and the KS0 25 probability of each KS0 vertex fit has to be greater than 106 . Each KS0 candidate must have þ invariant mass within 12:1 MeV=c2 of the nominal K0 mass [17], and decay length with respect to the B vertex between 0.22 and 45 cm. The last criterion ensures that the decay vertices of the B0 and the KS0 are well separated. In addition, combinatorial background is suppressed by selecting events for which the angle between the momentum vector of each KS0 candidate and the vector connecting the beamspot and the KS0 vertex is smaller than 0.0185 radians. We ensure a good B vertex fit quality by requiring that the charged pions of at least one of the KS0 candidates have hits in the two inner layers of the vertex tracker. A B meson candidate is characterized kinematically by the energy-substituted mass mES qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs=2 þ p~ i p~ B Þ2 =E2i p2B and the energy difference pffiffiffi E EB 12 s, where ðEB ; p~ B Þ and ðEi ; p~ i Þ are the four-vectors in the laboratory frame of the B-candidate and the initial electron-positron system, respectively, and pB is the magnitude of p~ B . The asterisk denotes the ð4SÞ frame, and s is the square of the invariant mass of the electron-positron system. We require 5:27 < mES < 5:29 GeV=c2 and jEj < 0:1 GeV. Following the calculation of these kinematic variables, each of the B candidates is refitted with its mass constrained to the world average value of the B meson mass [17] in order to improve the DP position resolution, and ensure that Eq. (4) holds. The sideband used for background studies is in the range 5:20 < mES < 5:27 GeV=c2 and jEj < 0:1 GeV. Backgrounds arise primarily from random combinations in continuum eþ e ! qq events (q ¼ d, u, s, c). To enhance discrimination between signal and continuum background, we use a neural network (NN) [19] to combine four discriminating variables: the angles with respect 0.9 100 0.8 0.7 80 0.6 hmax 15 120 1 cos(θmin) = 0.00 cos(θmin) = 0.25 cos(θmin) = 0.50 cos(θmin) = 0.75 cos(θmin) = 1.00 cos(θmax) = 0.00 cos(θmax) = 0.25 cos(θmax) = 0.50 cos(θmax) = 0.75 cos(θmax) = 1.00 20 smin [GeV 2/c 4] PHYSICAL REVIEW D 85, 054023 (2012) 2 10 0.5 60 0.4 40 0.3 5 0.2 20 0.1 0 0 2 4 6 8 10 12 14 16 18 20 22 24 smax [GeV2/c4] 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 hmin FIG. 1 (color online). Lines of constant helicity angle in the Dalitz plot of smin versus smax (left), and the magnitude of the Jacobian (gray scale on the right) mapping ðsmin ; smax Þ to ðhmin ; hmax Þ. For the latter see Eq. (20). 054023-7 J. P. LEES et al. PHYSICAL REVIEW D 85, 054023 (2012) amplitude analysis uses the variables mES and E, the NN output, and the SDP variables to discriminate signal from background. The selected on-resonance data sample is assumed to consist of signal and continuum background. The feed-through from B decays other than the signal is found to be negligible. Misreconstructed signal events are not considered as a separate event species, but are taken into account as a part of the signal. The likelihood function Li for event i is the sum X (21) L i ¼ Nj P ij ðmES ; E; NN; hmin ; hmax Þ; to the beam axis of the B momentum and B thrust axis in the ð4SÞ frame, and the zeroth- and second-order monomials L0;2 of the energy flow about P the B thrust axis. The monomials are defined by Ln ¼ i pi j cosi jn , where i is the angle with respect to the B thrust axis of track or neutral cluster i and pi is the magnitude of its momentum. The sum excludes the B candidate and all quantities are calculated in the ð4SÞ frame. The NN is trained with offresonance data, sideband data, and simulated signal events that pass the selection criteria. Approximately 0.5% of events passing the full selection have more than one candidate. When this occurs, we select the candidate for which the error-weighted average of the masses of the KS0 candidates is closest to the world average KS0 mass [17]. With the above selection criteria, we obtain a signal reconstruction efficiency of 6.6% that has been determined from a signal Monte Carlo (MC) sample generated using the same DP model and parameters as obtained from the data fit results. We estimate from this MC that 1.4% of the selected signal events are misreconstructed, and assign a systematic uncertainty (see Sec. III F). We use MC events to study the background from other B decays (B background). We expect fewer than 6 such events in our data sample. As these events are wrongly reconstructed, the mES and E distributions are continuumlike and as a result the events are mostly absorbed in the continuum background category. We assign a systematic uncertainty for B background contamination in the signal. where j stands for the species (signal, continuum background) and Nj is the corresponding yield. Each probability density function (PDF) P ij is the product of four individual PDFs: D. The maximum-likelihood fit 1. The mES , E, and NN PDFs We perform an unbinned extended maximum-likelihood fit to extract the B0 ! KS0 KS0 KS0 event yield, as well as the resonant and nonresonant amplitudes. The fit for the The mES and E distributions of signal events are parametrized by an asymmetric Gaussian with power-law tails: Cr ðx; m0 ; l ; r ; l ; r Þ ¼ exp j P ij ¼ P ij ðmES ÞP ij ðEÞP ij ðNNÞP ij ðhmin ; hmax Þ: A study with fully reconstructed MC samples shows that correlations between the PDF variables are small and therefore we neglect them. However, possible small discrepancies in the fit results due to these correlations are accounted for in the systematic uncertainty (see Sec. III F). The total likelihood is given by X Y L ¼ exp Nj Li : (23) j ðx m0 Þ2 22i þ i ðx m0 Þ2 The m0 parameters for both mES and E are free in the fit to data, while the other parameters are fixed to values determined from a fit to MC simulation. For the NN distributions of signal we use a histogram PDF from MC simulation. For continuum events the mES and E PDFs are parametrized by an ARGUS shape function [20] and a straight line, respectively. The NN PDF is described by a sum of power functions: (22) x m0 < 0: x m0 0: i i¼l i ¼ r: (24) where x ¼ ðNN NNmin Þ=ðNNmax NNmin Þ and the N are normalization factors, computed analytically using the standard function, ð þ 2 þ Þ : (26) ð þ 1Þð þ 1Þ The parameters for all the continuum PDFs are determined by a fit to sideband data and then fixed for the fit in the signal region. N ð; Þ ¼ 2. Dalitz-plot PDFs Eðx; c1 ; a; b0 ; b1 ; b2 ; b3 ; c2 ; c3 Þ ¼ cos ðc1 Þ½cos ðaÞN ðb0 ; b1 Þx ð1 xÞ 2 2 b0 b1 þ sin2 ðaÞN ðb2 ; b3 Þxb2 ð1 xÞb3 þ sin2 ðc1 ÞN ðc2 ; c3 Þxc2 ð1 xÞc3 ; (25) The SDP PDF for continuum background is a histogram obtained from mES sideband on-resonance events. The SDP signal PDFs require as input the DP-dependent selection efficiency, " ¼ "ðhmin ; hmax Þ, that is described by a histogram and is taken from MC simulation. For each event we define the SDP signal PDF: 054023-8 AMPLITUDE ANALYSIS AND MEASUREMENT OF THE . . . PHYSICAL REVIEW D 85, 054023 (2012) FIG. 2. Two-dimensional scans of 2 lnL (gray scale) as a function of the mass and the width of an additional resonance. These scans were performed to look for an additional scalar resonance (left) and an additional tensor resonance (right). The baseline model of the scans for additional scalar resonances contains f0 ð980Þ, c0 , and NR intermediate states. The baseline model of the scans for additional tensor resonances contains f0 ð980Þ, c0 , NR, and f0 ð1710Þ intermediate states. The ellipses indicate the world average parameters [17] for the f0 ð1710Þ and f2 ð2010Þ resonances that are added to the model. P i sig ðhmin ; hmax Þ / "ðhmin ; hmax ÞjAðhmin ; hmax Þj2 : (27) The normalization of the PDF is implemented by numerical integration. To describe the experimental resolution in the SDP variables, we use an ensemble of two-dimensional histograms that represents the probability to reconstruct at the coordinate (hmin 0 , hmax 0 ) an event that has the true coordinate ðhmin ; hmax Þ. These histograms are taken from MC simulation and are convolved with the signal PDF. 3. Determination of the signal Dalitz-plot model Using on-resonance data, we determine a nominal signal DP model by making likelihood scans with various combinations of isobars. We start from a baseline model that includes f0 ð980Þ, c0 , and NR components. We then add another scalar resonance described by the RBW parametrization. We scan the likelihood by fixing the width and 10 mass of this additional resonance at several consecutive values, for each of which the fit to the data is repeated. All isobar magnitudes and phases are floating in these fits. From the scans we observe a significant improvement of the fit around a width and mass that are compatible with the values of the f0 ð1710Þ resonance [17]. After adding the f0 ð1710Þ to the nominal model we repeat the same procedure for an additional tensor particle. We find that the f2 ð2010Þ has a significant contribution. The results of the likelihood scans are shown in Fig. 2 in terms of 2 lnL ¼ 2 lnL ð2 lnLÞmin , where ð2 lnLÞmin corresponds to the minimal value obtained in the particular scan. To conclude the search for possible resonant contributions we add all well established resonances [17] and check if the likelihood increases. We do not find any other significant resonant contribution, but as we cannot exclude small contributions from the f0 ð1370Þ, f2 ð1270Þ, f20 ð1525Þ, 120 9 70 60 0.7 7 80 6 60 5 hmin smin [GeV 2/c 4] 80 0.8 100 8 1 0.9 0.6 50 0.5 40 0.4 4 40 30 0.3 20 3 20 2 0.2 10 0.1 1 10 12 14 16 18 20 22 24 0 smax [GeV2/c4] 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 hmax FIG. 3. Symmetrized (left) and square (right) DP for MC simulated signal events using the amplitudes obtained from the fit to data. The low population in bins along the edge of the symmetrized DP is due to the fact that the phase-space boundaries do not coincide with the histogram bin boundaries. 054023-9 PHYSICAL REVIEW D 85, 054023 (2012) 30 35 Weight/(0.001 GeV/c2) Weight/(0.001 GeV/c2) J. P. LEES et al. 30 25 20 15 10 5 25 20 15 10 5 Norm. Residuals 0 2 0 -2 2 0 -2 5.27 5.272 5.274 5.276 5.278 5.28 5.282 5.284 5.286 5.288 5.29 5.27 5.272 5.274 5.276 5.278 5.28 5.282 5.284 5.286 5.288 5.29 mES [GeV/c2] mES [GeV/c2] 60 35 50 30 Weight/(0.01 GeV) Weight/(0.01 GeV) Norm. Residuals 0 40 30 20 10 25 20 15 10 5 Norm. Residuals Norm. Residuals 0 2 0 -2 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 2 0 -2 0.1 -0.1 -0.08 -0.06 -0.04 -0.02 ∆E [GeV] 0 0.02 0.04 0.06 0.08 0.1 0.7 0.8 0.9 1 ∆E [GeV] 50 Weight/(0.05) 40 30 20 30 20 10 10 0 0 Norm. Residuals Norm. Residuals Weight/(0.05) 40 2 0 -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 0 -2 0 NN 0.1 0.2 0.3 0.4 0.5 0.6 NN FIG. 4 (color online). s P lots (points with error bars) and PDFs (histograms) of the discriminating variables: mES (top), E (middle), and NN (bottom), for signal events (left) and continuum events (right). Below each bin are shown the residuals, normalized in error units. The horizontal dotted and full lines mark the one and two standard deviation levels, respectively. a0 ð1450Þ, and f0 ð1500Þ resonances, we assign model uncertainties (see Sec. III F) due to not taking these resonances into account. E. Results The maximum-likelihood fit of 505 candidates results in a B0 ! KS0 KS0 KS0 event yield of 200 15 and a continuum yield of 305 18, where the uncertainties are statistical only. The symmetrized and square Dalitz plots of a signal DP-model MC sample generated with the result of the fit to data are shown in Fig. 3. Figure 4 shows plots of E, mES , and the NN for isolated signal and continuum background events obtained by the s P lots [21] technique. Figure 5 shows projections of the data onto the invariant masses smin and smax . When the fit is repeated with initial parameter values randomly chosen within wide ranges above and below the nominal values for the magnitudes and within the ½; interval for the phases, we observe convergence towards two solutions with minimum values of the negative 054023-10 PHYSICAL REVIEW D 85, 054023 (2012) Events/(0.0825 GeV/c2) 80 70 60 50 40 30 20 10 Norm. Residuals Norm. Residuals Events/(0.115 GeV/c2) AMPLITUDE ANALYSIS AND MEASUREMENT OF THE . . . 2 0 -2 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 80 70 60 50 40 30 20 10 2 0 -2 3.2 3.4 3.6 2 3.8 4 4.2 4.4 4.6 4.8 2 smin [GeV/c ] smax [GeV/c ] pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi FIG. 5 (color online). Projections onto smin (left) and smax (right). On-resonance data are shown as points with error bars while the dashed (dotted) histogram represents the signal (continuum) component. The solid-line histogram is the total PDF. Below each bin are shown the residuals, normalized in error units. The horizontal dotted and full lines mark the one and two standard deviation levels, respectively. log-likelihood function 2 lnL separated by 3.25 units. In the following, we refer to them as Solution 1 (the global minimum) and Solution 2 (a local minimum). No other local minima were found. In the fit, we measure directly the relative magnitudes and phases of the different components of the signal model. The magnitude and phase of the NR amplitude are fixed to 1 and 0, respectively, as a reference. In Fig. 6 we show likelihood scans of the isobar magnitudes and phases of all the resonances, where both solutions can be noticed. Each of these scans is obtained by fixing the corresponding isobar parameter at several consecutive values, for each of which the fit to the data is repeated. The measured relative amplitudes c are used to extract the fit fraction (FF) defined as P3k P3k ¼3k2 ¼3k2 c c hF F i P ; (28) FF ðkÞ ¼ c c hF F i where k, which varies from 1 to 5, represents an intermediate state. Each fit fraction is a sum of three identical contributions, one for each pair of KS0 . The indices and run from 1 to 15, as each of the five resonances contributes to three pairs of KS0 , which correspond to the three terms (3k 2, 3k 1, and 3k) in each sum in the numerator of Eq. (28). The dynamical amplitudes F are defined in Sec. III A and the terms ZZ hF F i ¼ F F dsmin dsmax (29) are obtained by integration over the DP. The total fit fraction is defined as the algebraic sum of all fit fractions. This quantity is not necessarily unity due to the potential presence of net constructive or destructive interference. In order to estimate the statistical significance of each resonance, we evaluate the difference lnL between the log-likelihood of the nominal fit and that of a fit where the magnitude of the amplitude of the resonance is set to 0 (this difference can be directly read from the likelihood scans as a function of magnitudes in Fig. 6). In this case the phase of the resonance becomes meaningless, and we therefore account for 2 degrees of freedom removed from the fit. The value 2 lnL is used to evaluate the p-value for 2 degrees of freedom; we determine the equivalent onedimensional significance from this p-value. The results for the phase and the fit fraction are given in Table II for the two solutions; the change in likelihood when the amplitude of the resonance is set to 0 and the resulting statistical significance of each resonance is given for Solution 1. As the fit fractions are not parameters of the PDF itself, their statistical errors are obtained from the 68.3% coverage intervals of the fit-fraction distributions obtained from a large number of pseudoexperiments generated with the corresponding solution (1 or 2). As observed in other threekaon modes [6–10], the total FF significantly exceeds unity. In Table II it can be seen that the two solutions differ mostly in the fraction assigned to the NR and the f0 ð980Þ components. Solution 1 corresponds to a small FF of the f0 ð980Þ and a large value for the NR, and Solution 2 has a large f0 ð980Þ fraction and a smaller NR fraction. Other three-kaon modes [6–10] favor the behavior of Solution 1. Generalizing Eq. (28), we obtain the interference fractions among the intermediate decay modes k and j: P3k FF ðk; jÞ ¼ P3j ¼3j2 c c hF F i P ; c c hF F i ¼3k2 (30) which are given in Table III for Solution 1. Unlike the total FF defined above, the elements of this matrix sum to unity. The large destructive interference between the f0 ð980ÞKS0 and the NR components appears clearly in the table. This is possible due to the large overlap in phase space between 054023-11 20 18 16 14 12 10 8 6 4 2 0 0 PHYSICAL REVIEW D 85, 054023 (2012) -2 ∆ ln L -2 ∆ ln L J. P. LEES et al. 0.2 0.4 0.6 0.8 1 1.2 1.4 20 18 16 14 12 10 8 6 4 2 0 -3 -2 Mag f (980) [arbitrary units] -2 ∆ ln L -2 ∆ ln L 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 20 18 16 14 12 10 8 6 4 2 0 -3 -2 Mag f (1710) [arbitrary units] -2 ∆ ln L -2 ∆ ln L 0.002 0.004 20 18 16 14 12 10 8 6 4 2 0 -3 -2 Mag f (2010) [arbitrary units] -2 ∆ ln L -2 ∆ ln L 0.02 0.04 0.06 Mag χ c0 0.08 0.1 2 3 -1 0 1 2 3 -1 0 1 2 3 2 3 Phase f 2(2010) [rad] 2 20 18 16 14 12 10 8 6 4 2 0 0 1 Phase f 0(1710) [rad] 0 20 18 16 14 12 10 8 6 4 2 0 0 0 Phase f 0(980) [rad] 0 20 18 16 14 12 10 8 6 4 2 0 0 -1 0.12 0.14 [arbitrary units] 20 18 16 14 12 10 8 6 4 2 0 -3 -2 -1 0 1 Phase χ [rad] c0 FIG. 6 (color online). One-dimensional scans of 2 lnL as a function of magnitudes (left) and phases (right) of the resonances f0 ð980Þ, f0 ð1710Þ, f2 ð2010Þ, and c0 (top to bottom). The horizontal dashed lines mark the one and two standard deviation levels. 054023-12 AMPLITUDE ANALYSIS AND MEASUREMENT OF THE . . . TABLE II. Summary of measurements of the quasi-two-body parameters. The quoted uncertainties are statistical only. The change in the log-likelihood ( 2 lnL) corresponds to the case where the magnitude of the amplitude of the resonance is set to 0. This number is used for the estimation of the statistical significance of each resonance. Mode Parameter Solution 1 Solution 2 f0 ð980ÞKS0 FF Phase [rad] 2 lnL Significance [] 0:44þ0:20 0:19 0:09 0:16 11.7 3.0 1:03þ0:22 0:17 1:26 0:17 f0 ð1710ÞKS0 FF Phase [rad] 2 lnL Significance [] 0:07þ0:07 0:03 1:11 0:23 14.2 3.3 0:09þ0:05 0:02 0:36 0:20 f2 ð2010ÞKS0 FF Phase [rad] 2 lnL Significance [] 0:09þ0:03 0:03 2:50 0:20 14.0 3.3 0:10 0:02 1:58 0:22 NR FF Phase [rad] 2 lnL Significance [] 2:16þ0:36 0:37 0.0 68.1 8.0 1:37þ0:26 0:21 0.0 c0 KS0 FF Phase [rad] 2 lnL Significance [] 0:07þ0:04 0:02 0:63 0:47 18.5 3.9 0:07 0:02 0:24 0:52 Total FF 2:84þ0:71 0:66 2:66þ0:35 0:27 the exponential NR term and the broad tail of the f0 ð980Þ resonance above the KK threshold. Using the relative fit fractions, we calculate the branching fraction B for the intermediate mode k as FF ðkÞ BðB0 ! KS0 KS0 KS0 Þ; where BðB ! fraction: 0 KS0 KS0 KS0 Þ (31) is the total inclusive branching Nsig : (32) "N BB We estimate the average efficiency " ¼ 6:6% using a fully reconstructed DP-model MC sample generated with the parameters found in data. The results of the branching B ðB0 ! KS0 KS0 KS0 Þ ¼ PHYSICAL REVIEW D 85, 054023 (2012) fraction measurements are shown in Table IV. As a crosscheck we attempt to compare our measured branching fractions to results from other measurements; however, many of the branching fractions for the decay into kaons of the resonances included in our model are not (or are only poorly) measured (marked as ‘‘seen’’ in Ref. [17]). An exception is the charmonium state c0 , for which the measured value is Bðc0 ! KS0 KS0 Þ ¼ ð3:16 0:18Þ 103 [17]. We can then use the BABAR measurement of 6 BðB0 ! c0 K0 Þ ¼ ð142þ55 [22] 44 8 16 12Þ 10 1 0 0 0 0 to calculate B½B ! c0 ð! KS KS ÞKS ¼ 2 BðB0 ! c0 K0 Þ Bðc0 ! KS0 KS0 Þ ¼ ð0:224 0:078Þ 106 , which is consistent with our measured branching fraction, given in Table IV. An interesting conclusion from this first amplitude analysis of the B0 ! KS0 KS0 KS0 decay mode is that we do not need to include a broad scalar fX ð1500Þ resonance, as has been done in other measurements [6–10], to describe the data. The peak in the invariant mass between 1.5 and 1:6 GeV=c2 can be described by the interference between the f0 ð1710Þ resonance and the nonresonant component. However, minor contributions from the f20 ð1525Þ and f0 ð1500Þ resonances to this structure cannot be excluded. F. Systematic uncertainties Systematic effects are divided into model and experimental uncertainties. Details on how they have been estimated are given below and the associated numerical values are summarized in Table V. 1. Model uncertainties We vary the mass, width, and any other parameter of all isobar fit components within their errors, as quoted in Table I, and assign the observed differences in our observables as the first part of the model uncertainty (‘‘model’’ in Table V). To estimate the contribution to B0 ! KS0 KS0 KS0 from resonances that are not included in our signal model but cannot be excluded statistically, namely, the f0 ð1370Þ, f2 ð1270Þ, f20 ð1525Þ, a0 ð1450Þ, and f0 ð1500Þ resonances, we perform fits to pseudoexperiments that include these resonances. The masses and the widths are taken from [17], except for the f0 ð1370Þ for which we take the values from [23]. We generate pseudoexperiments with the additional TABLE III. The interference fractions FFðk; jÞ among the intermediate decay amplitudes for Solution 1. Note that the diagonal elements are those defined in Eq. (28) and detailed in Table II. The lower diagonal elements are omitted since the matrix is symmetric. f0 ð980ÞKS0 f0 ð1710ÞKS0 f2 ð2010ÞKS0 NR c0 KS0 f0 ð980ÞKS0 f0 ð1710ÞKS0 f2 ð2010ÞKS0 NR c0 KS0 0.44 0.07 0.07 0:02 0:01 0.09 0:80 0:17 0.02 2.16 0.01 0:0003 0.0002 0:02 0.07 054023-13 J. P. LEES et al. PHYSICAL REVIEW D 85, 054023 (2012) TABLE IV. Summary of measurements of branching fractions (B). The quoted numbers are obtained by multiplying the corresponding fit fraction from Solution 1 by the measured inclusive B0 ! KS0 KS0 KS0 branching fraction. The first uncertainty is statistical, the second is systematic, and the third represents the signal DP-model dependence. B [ 106 ] Mode Inclusive B0 ! KS0 KS0 KS0 f0 ð980ÞKS0 , f0 ð980Þ ! KS0 KS0 f0 ð1710ÞKS0 , f0 ð1710Þ ! KS0 KS0 f2 ð2010ÞKS0 , f2 ð2010Þ ! KS0 KS0 NR, KS0 KS0 KS0 c0 KS0 , c0 ! KS0 KS0 6:19 0:48 0:15 0:12 2:7þ1:3 1:2 0:4 1:2 0:50þ0:46 0:24 0:04 0:10 0:54þ0:21 0:20 0:03 0:52 13:3þ2:2 2:3 0:6 2:1 0:46þ0:25 0:17 0:02 0:21 resonances, where the isobar magnitudes and phases have been determined in fits to data, and fit these data sets with the nominal model. We assign the induced shift in the observables as a second part of the model uncertainty. systematic uncertainty (‘‘B background’’ in Table V). We assign a systematic uncertainty for all fixed PDF parameters by varying them within their uncertainties according to the covariance matrix. We vary the histogram PDFs, i.e., the SDP PDF for continuum and the NN PDF for signal (‘‘discriminating variables’’ in Table V). The mES dependence of the SDP PDF for continuum was found to be negligible. We account for differences between simulation and data observed in the control sample B0 ! J= c KS0 (‘‘MC data’’ in Table V). These differences were estimated by propagating the differences, in the control sample, between backgroundsubtracted data and signal MC, into the fit PDFs. For the branching fraction measurement, we assign a systematic uncertainty due to the error on the calculation of NBB (‘‘NBB ’’ in Table V) and to the KS0 reconstruction efficiency. We correct the KS0 reconstruction efficiency by the difference between the efficiency found in a dedicated KS0 data sample and that found in simulation. We assign the uncertainty on the correction as a systematic error (‘‘KS0 reconstruction’’ in Table V). 2. Experimental systematic uncertainties To validate the analysis procedure, we perform fits on a large number of pseudoexperiments generated with the measured yields of signal events and continuum background. The signal events are taken from fully reconstructed MC that has been generated with the fit result to data. We observe small biases in the isobar magnitudes and phases. We correct for these biases by shifting the values of the parameters and assign to this procedure a systematic uncertainty, which corresponds to half the correction combined in quadrature with its error. This uncertainty accounts also for correlations between the signal variables, wrongly reconstructed events, and effects due to the limited sample size (‘‘fit bias’’ in Table V). From MC we estimate that there are six B background events in our data sample. To determine the bias introduced by these events, we add B background events from MC to our data sample, and fit it with the nominal model. We then assign the observed differences in the observables as a IV. TIME-DEPENDENT ANALYSIS In Sec. IVA we describe the proper-time distribution used to extract the time-dependent CP asymmetries. In Sec. IV B we explain the selection requirements used to obtain the signal candidates and suppress backgrounds. In Sec. IV C we describe the fit method and the approach used to account for experimental effects. In Sec. IV D we present the results of the fit, and finally, in Sec. IV E we discuss systematic uncertainties in the results. A. Proper-time distribution The time-dependent CP asymmetries are functions of the proper-time difference t ¼ tCP ttag between a fully reconstructed B0 ! KS0 KS0 KS0 decay (BCP ) and the other B meson decay in the event (Btag ), which is partially reconstructed. The observed decay rate is the physical decay rate modified to include tagging imperfections, namely, hDic TABLE V. Summary of systematic uncertainties. The model uncertainty is dominated by the variation of the line shapes due to the contribution of the poorly measured f2 ð2010Þ. Parameter Fit bias B background Discriminating variables MC data NBB KS0 reconstruction Sum Model 0 0 0 0 6 0.030 0.053 0.015 0.067 0.111 0.145 0.120 BðB ! KS KS KS Þ½10 0.011 FF f0 ð980Þ FF f0 ð1710Þ FF f2 ð2010Þ FF NR FF c0 Phase [rad] f0 ð980Þ Phase [rad] f0 ð1710Þ Phase [rad] f2 ð2010Þ Phase [rad] c0 0.013 0.007 0.005 0.024 0.002 0.008 0.011 0.044 0.039 0.056 0.001 0.001 0.083 0.000 0.018 0.020 0.014 0.011 0.006 0.001 0.003 0.023 0.001 0.014 0.001 0.004 0.010 054023-14 0.001 0.001 0.001 0.001 0.000 0.000 0.003 0.002 0.007 0.058 0.007 0.006 0.090 0.002 0.024 0.023 0.046 0.042 0.190 0.016 0.084 0.344 0.034 0.177 0.185 0.684 0.498 AMPLITUDE ANALYSIS AND MEASUREMENT OF THE . . . and Dc ; the former is the rate of correctly assigning the flavor of the B meson, averaged over B0 and B 0 , and the latter is the difference between Dc for B0 and B 0 . The index c denotes different quality categories of the tag-flavor P i sig ðt; t ; qtag ; cÞ ¼ PHYSICAL REVIEW D 85, 054023 (2012) assignment. Furthermore the decay rate is convolved with the per-event t resolution Rsig ðt; t Þ, which is described by the sum of three Gaussians and depends on t and its error t . For an event i with tag flavor qtag , one has ejtj= B0 Dc þ qtag hDic ½S sinðmd tÞ C cosðmd tÞ Rsig ðt; t Þ; 1 þ qtag 4 B0 2 where qtag is defined to be þ1 ( 1) for Btag ¼ B0 (Btag ¼ B 0 ), B0 is the mean B0 lifetime, and md is the mixing frequency [17]. The widths of the B0 and the B 0 are assumed to be the same. B. Event selection and backgrounds We reconstruct B0 ! KS0 KS0 KS0 candidates either from three KS0 ! þ candidates or from two KS0 ! þ and one KS0 ! 0 0 , where the 0 candidates are formed from pairs of photons. The vertex fit requirements are the same as in the amplitude analysis, and also the requirement that the charged pions of at least one of the KS0 have hits in the two inner layers of the vertex tracker. The KS0 candidates in the B0 ! 3KS0 ðþ Þ submode must have mass within 12 MeV=c2 of the nominal K 0 mass [17] and decay length with respect to the B vertex between 0.2 and 40 cm. In addition, combinatorial background is suppressed in both submodes by imposing that the angle between the momentum vector of each KS0 ðþ Þ candidate and the vector connecting the beamspot and the KS0 ðþ Þ vertex is smaller than 0.2 radians. Each KS0 decaying to charged pions in the B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ submode is required to have decay length between 0.15 and 60 cm and þ invariant mass less than 11 MeV from the world average KS0 mass [17]. The KS0 decaying to neutral pions in the B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ submode must have 0 0 invariant mass between 0:48 and 0:52 GeV=c2 . (33) Additionally, the neutral pions are selected if they have invariant mass between 0:100 and 0:141 GeV=c2 and if the photons have energies greater than 50 MeV in the laboratory frame and a lateral energy deposition profile in the electromagnetic calorimeter consistent with that expected for an electromagnetic shower (lateral moment [24] less than 0.55). The fact that we do not model any PDF using sideband data allows a loose requirement on mES and E in the time-dependent analysis, namely, 5:22 < mES < 5:29 GeV=c2 and 0:18 < E < 0:12 GeV. In case of multiple candidates passing the selection, we proceed in the same way as in the amplitude analysis. We use the same NN as in the amplitude analysis to suppress continuum background. With the above selection criteria, we obtain signal reconstruction efficiencies of 6.7% and 3.1% for the B0 ! 3KS0 ðþ Þ and B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ submodes, respectively. These efficiencies are determined from a DPmodel MC sample generated using the results of the amplitude analysis. We estimate from MC that 2.1% of the selected signal events are misreconstructed for B0 ! 3KS0 ðþ Þ, while the figure is 2.4% in B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ, and we do not treat these events differently from correctly reconstructed events. Because of the looser requirements, there are more background events from B decays than in the amplitude analysis, in particular, in the B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ submode. These backgrounds are included in the fit model and are summarized TABLE VI. Summary of B background modes included in the fit model of the time-dependent analysis. The expected number of events takes into account the branching fractions (B) and efficiencies. In case there is no measurement, the branching fraction of an isospin-related channel is used. All the fixed yields are varied by 100% for systematic uncertainties. Submode B0 ! 3KS0 ðþ Þ B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ Background mode Varied B [ 106 ] Number of events KS0 KS0 KL0 KS0 KS0 K0 KS0 KS0 K þ B0 ! fneutral generic decaysg Bþ ! fcharged generic decaysg No No No Yes Yes 2.4 27.5 11.5 Not applicable Not applicable 0.71 9.55 4.27 21.7 15.5 KS0 KS0 KL0 KS0 KS0 K0 KS0 KL0 K 0 KS0 KS0 K þ KS0 KS0 K þ 0 B ! fneutral generic decaysg Bþ ! fcharged generic decaysg No No No No No Yes Yes 2.4 27.5 27.5 11.5 27.5 Not applicable Not applicable 0.67 5.3 0.3 2.9 7.2 73.6 73.8 054023-15 J. P. LEES et al. PHYSICAL REVIEW D 85, 054023 (2012) in Table VI. As the analysis is phase-space integrated, we cannot model the c0 resonance separately, and its contribution to the CP asymmetries could cloud deviations in the charmless contributions. We therefore apply a veto around the invariant mass of this charmonium state. C. The maximum-likelihood fit We perform an unbinned extended maximum-likelihood fit to extract the B0 ! KS0 KS0 KS0 event yields along with the S and C parameters of the time-dependent analysis. The fit uses as variables mES , E, the NN output, t, and t . The selected on-resonance data sample is assumed to consist of signal, continuum background, and backgrounds from B decays. Wrongly reconstructed signal events are not considered separately. The likelihood function Li for event i is the sum Li ¼ X Nj P ij ðmES ; E; t; t ; NN; qtag ; c; pÞ; (34) j where j stands for the species (signal, continuum background, one for each B background category) and Nj is the corresponding yield; qtag , c, and p are the tag flavor, the tagging category, and the physics category, respectively. To determine qtag and c we use the B flavor-tagging algorithm of Ref. [25]. This algorithm combines several different signatures, such as charges, momenta, and decay angles of charged particles in the event to achieve optimal separation between the two B flavors. This produces six mutually exclusive tagging categories. We also retain untagged events in a seventh category; although these events do not contribute to the measurement of the timedependent CP asymmetry, they do provide additional sensitivity for the measurement of direct CP violation [26]. The two physics categories correspond to B0 ! 3KS0 ðþ Þ and B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ decays. The PDF for species j evaluated for event i is given by the product of individual PDFs: P ij ðmES ;E;t;t ;NN;qtag ;c;pÞ ¼ P ij ðmES ;pÞP ij ðE;pÞP ij ðNN;c;pÞP ij ðt;t ;qtag ;c;pÞ: (35) To take into account the different reconstruction of the two submodes, we use separate PDFs for the two physics categories. Separate NN and t PDFs are included for each tagging category within each physics category. The separate t PDFs for the two physics categories allow us to fit the S and C parameters either separately for the two submodes, or together. The total likelihood is given by X Y L ¼ exp Nj Li : j 1. t PDFs The signal PDF for t is given in Eq. (33). Parameters that depend solely on the tag side of the events (namely, ðÞ hDic and Dc ) are taken from the analysis of B ! ccK decays [27]. On the other hand, parameters that depend on the signal-side reconstruction, due to the absence of direct tracks from the B decay, cannot be taken from modes that include such direct tracks. This is the case for the parameters that describe the resolution function, which are found in a fit to simulated events. A systematic uncertainty for data-MC differences is assigned using the control sample B0 ! J= c KS0 , as explained in Sec. IV E. For continuum events we use a zero-lifetime component. This parametrization is convolved with the same resolution function as for signal, with different parameters that are varied in the fit to data. The parameters of this PDF are not separated in the tagging categories. The small contribution from eþ e ! cc events is well described by the tails of the resolution function. For B background events we use the signal PDF, with resolution parameters from the BABAR ðÞ analysis [27]. The parameters S and C are set B ! ccK to zero and varied to assign a systematic uncertainty. 2. Description of the other variables The mES and E distributions of signal events are parametrized by an asymmetric Gaussian with power-law tails, as given in Eq. (24), and, for mES , a small additional component, parametrized by an ARGUS shape function [20], to correctly describe misreconstructed events. The means in these two PDFs for B0 ! 3KS0 ðþ Þ events are allowed to vary in the fit to data, and the other parameters are taken from MC simulation. For the NN distributions of signal we use histogram PDFs taken from MC simulation for each physics and tagging category. The mES , E, and NN PDFs for continuum events are parametrized by an ARGUS shape function, a straight line, and the sum of power functions from Eq. (25), respectively. All continuum parameters, except for c2 and c3 of the NN PDF, are allowed to vary in the fit. All the fixed parameters are varied, within the uncertainties found in a fit to sideband data, to estimate systematic errors. All the B background PDFs are described by fixed histograms taken from MC simulation. D. Results The maximum-likelihood fit of 3261 candidates in the B0 ! 3KS0 ðþ Þ submode and 7209 candidates in the B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ submode results in the event yields detailed in Table VII. The fit result for the time-dependent CP-violation parameters S and C is (36) i 054023-16 S ¼ 0:94þ0:24 0:21 ; C ¼ 0:17 0:18; AMPLITUDE ANALYSIS AND MEASUREMENT OF THE . . . TABLE VII. Event yields for the different event species, resulting from the maximum-likelihood fit for the time-dependent analysis. ‘‘Bþ B (B0 B 0 ) background’’ represents background from charged (neutral) B decays. Quoted uncertainties are statistical only. Species 3KS0 ðþ Þ 2KS0 ðþ ÞKS0 ð0 0 Þ 201þ16 15 3086þ56 54 54þ29 24 9þ31 30 62þ13 12 7086þ85 83 45þ34 30 4þ38 29 45 40 35 30 25 20 15 10 5 0 -8 where the uncertainties are statistical only. The correlation between S and C is 0:16. We use the fit result to create s P lots of the signal distributions of t, the timedependent asymmetry, and the discriminating variables. Figure 7 shows the t s P lots for the combined fit result and for the individual submodes. Figure 8 shows the signal distributions and Fig. 9 the continuum background distributions of the discriminating variables. The distributions shown in these three figures illustrate the good agreement between the data and the fit model. We scan the statistical-only likelihood of the S parameter for both submodes and for the combined fit. The result, on the left-hand side of Fig. 10, shows a sizable difference between the S values for the two submodes; the level of 1.5 1 Asymmetry Weight/(2 ps) Signal Continuum Bþ B background B0 B 0 background PHYSICAL REVIEW D 85, 054023 (2012) 0.5 0 -0.5 -1 -6 -4 -2 0 2 4 6 -1.5 -8 8 -6 -4 -2 ∆ t [ps] 0 2 4 6 8 2 4 6 8 2 4 6 8 ∆ t [ps] 1 Asymmetry Weight/(2 ps) 1.5 16 14 12 10 8 6 4 2 0 -2 -8 0.5 0 -0.5 -1 -6 -4 -2 0 2 4 6 -1.5 -8 8 -6 -4 -2 45 40 35 30 25 20 15 10 5 0 -8 0 ∆ t [ps] 1.5 1 Asymmetry Weight/(2 ps) ∆ t [ps] 0.5 0 -0.5 -1 -6 -4 -2 0 2 4 6 8 ∆ t [ps] -1.5 -8 -6 -4 -2 0 ∆ t [ps] FIG. 7 (color online). Signal s P lots (points with error bars) and PDFs (histograms) of t (left) and the derived asymmetry (right) for the B0 ! 3KS0 ðþ Þ submode (top), the B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ submode (middle), and the combined fit (bottom). In the t distributions on the left-hand side, points marked with and solid lines correspond to decays where Btag is a B0 meson; points marked with and dashed lines correspond to decays where Btag is a B 0 meson. Points of the asymmetry s P lots that are outside the range of a figure are marked by arrows. 054023-17 PHYSICAL REVIEW D 85, 054023 (2012) Weight/(0.0028 GeV/c2) 30 80 60 40 20 0 2 0 -2 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 25 20 15 10 5 0 -5 Norm. Residuals Norm. Residuals Weight/(0.0028 GeV/c2) J. P. LEES et al. 2 0 -2 5.22 5.23 5.24 2 0 -2 -0.1 -0.05 0 0.05 0.1 -0.15 5.28 5.29 -0.1 -0.05 0 0.05 0.1 ∆E [GeV] 15 Weight/(0.05) 40 30 20 10 10 5 0 -5 -10 0 2 0 -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Norm. Residuals Weight/(0.04) 5.27 2 0 -2 ∆E [GeV] Norm. Residuals 5.26 16 14 12 10 8 6 4 2 0 -2 Weight/(0.012 GeV) 70 60 50 40 30 20 10 0 -0.15 5.25 mES [GeV/c2] Norm. Residuals Norm. Residuals Weight/(0.012 GeV) mES [GeV/c2] NN 2 0 -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 NN FIG. 8 (color online). Signal s P lots (points with error bars) and PDFs (histograms) of the discriminating variables: mES (top), E (middle), and the NN output (bottom) for the B0 ! 3KS0 ðþ Þ submode (left) and for the B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ submode (right). Below each bin are shown the residuals, normalized in error units. The horizontal dotted and full lines mark the one and two standard deviation levels, respectively. consistency, conservatively estimated from the sum of the two individual likelihood scans, is approximately 2:6 (a p-value of 1.0% with 2 degrees of freedom). This value is obtained including only the dominant statistical uncertainty and neglecting the small correlation between the CP-violation parameters. The results obtained when S and C are allowed to vary individually for each of the þ0:17 0 submodes are S ¼ 1:42þ0:27 0:24 , C ¼ 0:140:17 for B ! 0 þ þ0:56 þ0:42 3KS ð Þ and S ¼ 0:400:57 , C ¼ 0:190:43 for B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ. In both cases the quoted uncertainties are statistical only. As there is some correlation between the S and C parameters, we perform a two-dimensional statistical likelihood scan of the combined likelihood, which is then convolved by the systematic uncertainties on S and C (systematic uncertainties are discussed in Sec. IV E). The result is shown on the right-hand side of Fig. 10. We find that CP conservation is excluded at 3.8 standard deviations, and thus, for the first time, we measure an evidence of CP violation in B0 ! KS0 KS0 KS0 decays. The ðÞ is difference between our result and that from B0 ! ccK less than 2 standard deviations. The scan also shows that the result is close to the physical boundary, given by the constraint S 2 þ C2 1. E. Systematic uncertainties The systematic uncertainties are summarized in Table VIII. The ‘‘MCstat ’’ uncertainty accounts for the limited size of the simulated data samples used to create the PDFs. The ‘‘Breco ’’ uncertainty propagates the experimental uncertainty in the measurement of tag-side-related quantities taken from [27] to our measurement. The ‘‘B background’’ contribution results from the uncertainty in the CP content and the branching fractions of fixed yields in the model of background from B decays. The dominant ‘‘MC data: t’’ systematic uncertainty is due to possible 054023-18 PHYSICAL REVIEW D 85, 054023 (2012) Weight/(0.0028 GeV/c2) 180 160 140 120 100 80 60 40 20 2 0 -2 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 Norm. Residuals Norm. Residuals Weight/(0.0028 GeV/c2) AMPLITUDE ANALYSIS AND MEASUREMENT OF THE . . . 350 300 250 200 150 100 50 2 0 -2 5.22 5.23 5.24 mES [GeV/c2] Weight/(0.012 GeV) Weight/(0.012 GeV) 180 160 140 120 100 Norm. Residuals Norm. Residuals 80 2 0 -2 -0.15 -0.1 -0.05 5.25 0 0.05 0.1 5.28 5.29 -0.15 -0.1 -0.05 0 0.05 0.1 ∆E [GeV] 300 600 250 500 Weight/(0.04) Weight/(0.04) 5.27 380 360 340 320 300 280 260 240 220 200 2 0 -2 ∆E [GeV] 200 150 100 400 300 200 100 2 0 -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Norm. Residuals 50 Norm. Residuals 5.26 mES [GeV/c2] 2 0 -2 0 0.1 0.2 0.3 0.4 NN 0.5 0.6 0.7 0.8 0.9 1 6 8 NN Weight/(0.64 ps) Weight/(0.64 ps) 250 200 150 100 2 0 -2 -8 400 300 200 100 -6 -4 -2 0 2 4 6 8 Norm. Residuals Norm. Residuals 50 500 ∆ t [ps] 2 0 -2 -8 -6 -4 -2 0 2 4 ∆ t [ps] FIG. 9 (color online). Continuum s P lots (points with error bars) and PDFs (histograms) of mES , E, the NN output, and t (top to bottom). Plots on the left-hand side correspond to the B0 ! 3KS0 ðþ Þ submode, and on the right-hand side to the B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ submode. In the t distributions, points marked with and solid lines correspond to decays where Btag is a B0 meson; points marked with and dashed lines correspond to decays where Btag is a B 0 meson. Below each bin are shown the residuals, normalized in error units. The horizontal dotted and full lines mark the one and two standard deviation levels, respectively. differences between data and simulation concerning the procedure used to obtain the signal B decay vertex from tracks originating from KS0 decays. We quantify this uncertainty using the control sample B0 ! J= c KS0 by comparing the difference between t values obtained with and without the J= c in data and simulation. We then propagate the observed differences and their uncertainties to the resolution function. We use this new resolution function to refit the data and obtain an estimate of the effect on S and C. We also use the samples B0 ! J= c KS0 ðþ Þ and B0 ! J= c KS0 ð0 0 Þ to estimate simulation-data differences for the other variables in the submodes B0 ! 3KS0 ðþ Þ and B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ, respectively. This contribution is referred to as ‘‘MC data: 054023-19 J. P. LEES et al. PHYSICAL REVIEW D 85, 054023 (2012) 30 5 0.5 4 20 0 3 C -2 ∆ ln L 25 15 2 10 -0.5 1 5 0 -2 -1.5 -1 -0.5 S 0 0.5 -1.5 1 -1 -0.5 0 0 S FIG. 10 (color online). One-dimensional statistical scan of 2 lnL as a function of S (left) and the two-dimensional scan, including systematic uncertainty, as a function of S and C (right). In the left-hand plot, red points marked with correspond to the 0 ð0 0 Þ submode, and black points marked with B0 ! 3KS0 ðþ Þ submode, blue points marked with to the B0 ! 2KS0 ðþ ÞK pSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi to the combined fit. In the right-hand plot, the gray scale is given in units of 2 lnL. The result of the BABAR analyses of ðÞ decays [27] is indicated as a white ellipse and the physical boundary (S 2 þ C2 1) is marked as a gray line. The scan B0 ! ccK appears to be trimmed on the lower left since the PDF becomes negative outside the physical region (i.e., the white region does not indicate that the scan flattens out at 5). discriminating variables’’ in Table VIII. The ‘‘fit bias’’ uncertainty is evaluated using fits to fully reconstructed MC samples. It accounts for effects from wrongly reconstructed events and correlations between fit variables. The ‘‘vetoes’’ uncertainty is related to the veto on the invariant mass. It is evaluated using events that pass the veto in pseudoexperiments studies. Finally, the ‘‘miscellaneous’’ uncertainty includes contributions from doubly Cabibbosuppressed decays, silicon vertex tracker alignment, and the uncertainties in the boost of the ð4SÞ. These ðÞ contributions are taken from the BABAR B ! ccK analysis [27]. be from f0 ð980Þ, f0 ð1710Þ, f2 ð2010Þ, and a nonresonant component, and measured the individual fit fractions and phases of each component. We do not observe any significant contribution from the so-called fX ð1500Þ resonance seen in, for example, Bþ ! Kþ K Kþ [6]. The peak in the invariant mass between 1.5 and 1:6 GeV=c2 can be described by the interference between the f0 ð1710Þ resonance and the nonresonant component. We see some hints from the f20 ð1525Þ and f0 ð1500Þ resonances that could also contribute to this structure, but due to limited sample size we cannot make a significant statement. Future investigations of the KK system could shed more light on the situation. Furthermore we have performed an update of the phase-space-integrated time-dependent analysis of the same decay mode, using B0 ! 3KS0 ðþ Þ and B0 ! 2KS0 ðþ ÞKS0 ð0 0 Þ decays, with the final BABAR data set. We measure the CP-violation parameters to be S ¼ 0:94þ0:24 0:21 0:06 and C ¼ 0:17 0:18 0:04, where the first quoted uncertainty is statistical and the second is systematic. These measured values are consistent with and supersede those reported in Ref. [3]. They are compatible within two standard deviations with those measured in tree-dominated modes such as B0 ! J= c KS0 , as expected in the SM. For the first time, we report evidence of CP violation in B0 ! KS0 KS0 KS0 decays; CP conservation is excluded at 3.8 standard deviations including systematic uncertainties. V. SUMMARY ACKNOWLEDGMENTS We have performed the first amplitude analysis of B0 ! events and measured the total inclusive branching fraction to be ð6:19 0:48 0:15 0:12Þ 106 , where the first uncertainty is statistical, the second is systematic, and the third represents the signal DP-model dependence. We have identified the dominant contributions to the DP to We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminosity and machine conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the computing organizations that support BABAR. The collaborating institutions TABLE VIII. Summary of systematic uncertainties on the S and C parameters. S C MCstat Breco B background MC data: t MC data: discriminating variables Fit bias Vetoes Miscellaneous 0.002 0.004 0.032 0.045 0.021 0.022 0.006 0.004 0.001 0.003 0.012 0.027 0.004 0.018 0.004 0.015 Sum 0.064 0.038 Source KS0 KS0 KS0 054023-20 AMPLITUDE ANALYSIS AND MEASUREMENT OF THE . . . PHYSICAL REVIEW D 85, 054023 (2012) wish to thank SLAC for its support and the kind hospitality extended to them. This work is supported by the U.S. Department of Energy and National Science Foundation, the Natural Sciences and Engineering Research Council (Canada), the Commissariat à l’Energie Atomique and Institut National de Physique Nucléaire et de Physique des Particules (France), the Bundesministerium für Bildung und Forschung and Deutsche Forschungsgemeinschaft (Germany), the Istituto Nazionale di Fisica Nucleare (Italy), the Foundation for Fundamental Research on Matter (The Netherlands), the Research Council of Norway, the Ministry of Education and Science of the Russian Federation, Ministerio de Ciencia e Innovación (Spain), and the Science and Technology Facilities Council (United Kingdom). Individuals have received support from the Marie-Curie IEF program (European Union), the A. P. Sloan Foundation (USA), and the Binational Science Foundation (USA-Israel). [1] Y. Grossman and M. P. Worah, Phys. Lett. B 395, 241 (1997). [2] H.-Y. Cheng, C.-K. Chua, and A. Soni, Phys. Rev. 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